"""
=======
Cliques
=======
Find and manipulate cliques of graphs.
Note that finding the largest clique of a graph has been
shown to be an NP-complete problem; the algorithms here
could take a long time to run.
http://en.wikipedia.org/wiki/Clique_problem
"""
# Copyright (C) 2004-2008 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
import networkx
from networkx.utils.decorators import *
__author__ = """Dan Schult (dschult@colgate.edu)"""
__all__ = ['find_cliques', 'find_cliques_recursive', 'make_max_clique_graph',
'make_clique_bipartite' ,'graph_clique_number',
'graph_number_of_cliques', 'node_clique_number',
'number_of_cliques', 'cliques_containing_node',
'project_down', 'project_up']
@not_implemented_for('directed')
[docs]def find_cliques(G):
"""Search for all maximal cliques in a graph.
Maximal cliques are the largest complete subgraph containing
a given node. The largest maximal clique is sometimes called
the maximum clique.
Returns
-------
generator of lists: genetor of member list for each maximal clique
See Also
--------
find_cliques_recursive :
A recursive version of the same algorithm
Notes
-----
To obtain a list of cliques, use list(find_cliques(G)).
Based on the algorithm published by Bron & Kerbosch (1973) [1]_
as adapated by Tomita, Tanaka and Takahashi (2006) [2]_
and discussed in Cazals and Karande (2008) [3]_.
The method essentially unrolls the recursion used in
the references to avoid issues of recursion stack depth.
This algorithm is not suitable for directed graphs.
This algorithm ignores self-loops and parallel edges as
clique is not conventionally defined with such edges.
There are often many cliques in graphs. This algorithm can
run out of memory for large graphs.
References
----------
.. [1] Bron, C. and Kerbosch, J. 1973.
Algorithm 457: finding all cliques of an undirected graph.
Commun. ACM 16, 9 (Sep. 1973), 575-577.
http://portal.acm.org/citation.cfm?doid=362342.362367
.. [2] Etsuji Tomita, Akira Tanaka, Haruhisa Takahashi,
The worst-case time complexity for generating all maximal
cliques and computational experiments,
Theoretical Computer Science, Volume 363, Issue 1,
Computing and Combinatorics,
10th Annual International Conference on
Computing and Combinatorics (COCOON 2004), 25 October 2006, Pages 28-42
http://dx.doi.org/10.1016/j.tcs.2006.06.015
.. [3] F. Cazals, C. Karande,
A note on the problem of reporting maximal cliques,
Theoretical Computer Science,
Volume 407, Issues 1-3, 6 November 2008, Pages 564-568,
http://dx.doi.org/10.1016/j.tcs.2008.05.010
"""
# Cache nbrs and find first pivot (highest degree)
maxconn=-1
nnbrs={}
pivotnbrs=set() # handle empty graph
for n,nbrs in G.adjacency_iter():
nbrs=set(nbrs)
nbrs.discard(n)
conn = len(nbrs)
if conn > maxconn:
nnbrs[n] = pivotnbrs = nbrs
maxconn = conn
else:
nnbrs[n] = nbrs
# Initial setup
cand=set(nnbrs)
smallcand = set(cand - pivotnbrs)
done=set()
stack=[]
clique_so_far=[]
# Start main loop
while smallcand or stack:
try:
# Any nodes left to check?
n=smallcand.pop()
except KeyError:
# back out clique_so_far
cand,done,smallcand = stack.pop()
clique_so_far.pop()
continue
# Add next node to clique
clique_so_far.append(n)
cand.remove(n)
done.add(n)
nn=nnbrs[n]
new_cand = cand & nn
new_done = done & nn
# check if we have more to search
if not new_cand:
if not new_done:
# Found a clique!
yield clique_so_far[:]
clique_so_far.pop()
continue
# Shortcut--only one node left!
if not new_done and len(new_cand)==1:
yield clique_so_far + list(new_cand)
clique_so_far.pop()
continue
# find pivot node (max connected in cand)
# look in done nodes first
numb_cand=len(new_cand)
maxconndone=-1
for n in new_done:
cn = new_cand & nnbrs[n]
conn=len(cn)
if conn > maxconndone:
pivotdonenbrs=cn
maxconndone=conn
if maxconndone==numb_cand:
break
# Shortcut--this part of tree already searched
if maxconndone == numb_cand:
clique_so_far.pop()
continue
# still finding pivot node
# look in cand nodes second
maxconn=-1
for n in new_cand:
cn = new_cand & nnbrs[n]
conn=len(cn)
if conn > maxconn:
pivotnbrs=cn
maxconn=conn
if maxconn == numb_cand-1:
break
# pivot node is max connected in cand from done or cand
if maxconndone > maxconn:
pivotnbrs = pivotdonenbrs
# save search status for later backout
stack.append( (cand, done, smallcand) )
cand=new_cand
done=new_done
smallcand = cand - pivotnbrs
def find_cliques_recursive(G):
"""Recursive search for all maximal cliques in a graph.
Maximal cliques are the largest complete subgraph containing
a given point. The largest maximal clique is sometimes called
the maximum clique.
Returns
-------
list of lists: list of members in each maximal clique
See Also
--------
find_cliques : An nonrecursive version of the same algorithm
Notes
-----
Based on the algorithm published by Bron & Kerbosch (1973) [1]_
as adapated by Tomita, Tanaka and Takahashi (2006) [2]_
and discussed in Cazals and Karande (2008) [3]_.
This implementation returns a list of lists each of
which contains the members of a maximal clique.
This algorithm ignores self-loops and parallel edges as
clique is not conventionally defined with such edges.
References
----------
.. [1] Bron, C. and Kerbosch, J. 1973.
Algorithm 457: finding all cliques of an undirected graph.
Commun. ACM 16, 9 (Sep. 1973), 575-577.
http://portal.acm.org/citation.cfm?doid=362342.362367
.. [2] Etsuji Tomita, Akira Tanaka, Haruhisa Takahashi,
The worst-case time complexity for generating all maximal
cliques and computational experiments,
Theoretical Computer Science, Volume 363, Issue 1,
Computing and Combinatorics,
10th Annual International Conference on
Computing and Combinatorics (COCOON 2004), 25 October 2006, Pages 28-42
http://dx.doi.org/10.1016/j.tcs.2006.06.015
.. [3] F. Cazals, C. Karande,
A note on the problem of reporting maximal cliques,
Theoretical Computer Science,
Volume 407, Issues 1-3, 6 November 2008, Pages 564-568,
http://dx.doi.org/10.1016/j.tcs.2008.05.010
"""
nnbrs={}
for n,nbrs in G.adjacency_iter():
nbrs=set(nbrs)
nbrs.discard(n)
nnbrs[n]=nbrs
if not nnbrs: return [] # empty graph
cand=set(nnbrs)
done=set()
clique_so_far=[]
cliques=[]
_extend(nnbrs,cand,done,clique_so_far,cliques)
return cliques
def _extend(nnbrs,cand,done,so_far,cliques):
# find pivot node (max connections in cand)
maxconn=-1
numb_cand=len(cand)
for n in done:
cn = cand & nnbrs[n]
conn=len(cn)
if conn > maxconn:
pivotnbrs=cn
maxconn=conn
if conn==numb_cand:
# All possible cliques already found
return
for n in cand:
cn = cand & nnbrs[n]
conn=len(cn)
if conn > maxconn:
pivotnbrs=cn
maxconn=conn
# Use pivot to reduce number of nodes to examine
smallercand = set(cand - pivotnbrs)
for n in smallercand:
cand.remove(n)
so_far.append(n)
nn=nnbrs[n]
new_cand=cand & nn
new_done=done & nn
if not new_cand and not new_done:
# Found the clique
cliques.append(so_far[:])
elif not new_done and len(new_cand) is 1:
# shortcut if only one node left
cliques.append(so_far+list(new_cand))
else:
_extend(nnbrs, new_cand, new_done, so_far, cliques)
done.add(so_far.pop())
[docs]def make_max_clique_graph(G,create_using=None,name=None):
""" Create the maximal clique graph of a graph.
Finds the maximal cliques and treats these as nodes.
The nodes are connected if they have common members in
the original graph. Theory has done a lot with clique
graphs, but I haven't seen much on maximal clique graphs.
Notes
-----
This should be the same as make_clique_bipartite followed
by project_up, but it saves all the intermediate steps.
"""
cliq=list(map(set,find_cliques(G)))
if create_using:
B=create_using
B.clear()
else:
B=networkx.Graph()
if name is not None:
B.name=name
for i,cl in enumerate(cliq):
B.add_node(i+1)
for j,other_cl in enumerate(cliq[:i]):
# if not cl.isdisjoint(other_cl): #Requires 2.6
intersect=cl & other_cl
if intersect: # Not empty
B.add_edge(i+1,j+1)
return B
[docs]def make_clique_bipartite(G,fpos=None,create_using=None,name=None):
"""Create a bipartite clique graph from a graph G.
Nodes of G are retained as the "bottom nodes" of B and
cliques of G become "top nodes" of B.
Edges are present if a bottom node belongs to the clique
represented by the top node.
Returns a Graph with additional attribute dict B.node_type
which is keyed by nodes to "Bottom" or "Top" appropriately.
if fpos is not None, a second additional attribute dict B.pos
is created to hold the position tuple of each node for viewing
the bipartite graph.
"""
cliq=list(find_cliques(G))
if create_using:
B=create_using
B.clear()
else:
B=networkx.Graph()
if name is not None:
B.name=name
B.add_nodes_from(G)
B.node_type={} # New Attribute for B
for n in B:
B.node_type[n]="Bottom"
if fpos:
B.pos={} # New Attribute for B
delta_cpos=1./len(cliq)
delta_ppos=1./G.order()
cpos=0.
ppos=0.
for i,cl in enumerate(cliq):
name= -i-1 # Top nodes get negative names
B.add_node(name)
B.node_type[name]="Top"
if fpos:
if name not in B.pos:
B.pos[name]=(0.2,cpos)
cpos +=delta_cpos
for v in cl:
B.add_edge(name,v)
if fpos is not None:
if v not in B.pos:
B.pos[v]=(0.8,ppos)
ppos +=delta_ppos
return B
def project_down(B,create_using=None,name=None):
"""Project a bipartite graph B down onto its "bottom nodes".
The nodes retain their names and are connected if they
share a common top node in the bipartite graph.
Returns a Graph.
"""
if create_using:
G=create_using
G.clear()
else:
G=networkx.Graph()
if name is not None:
G.name=name
for v,Bvnbrs in B.adjacency_iter():
if B.node_type[v]=="Bottom":
G.add_node(v)
for cv in Bvnbrs:
G.add_edges_from([(v,u) for u in B[cv] if u!=v])
return G
def project_up(B,create_using=None,name=None):
"""Project a bipartite graph B down onto its "bottom nodes".
The nodes retain their names and are connected if they
share a common Bottom Node in the Bipartite Graph.
Returns a Graph.
"""
if create_using:
G=create_using
G.clear()
else:
G=networkx.Graph()
if name is not None:
G.name=name
for v,Bvnbrs in B.adjacency_iter():
if B.node_type[v]=="Top":
vname= -v #Change sign of name for Top Nodes
G.add_node(vname)
for cv in Bvnbrs:
# Note: -u changes the name (not Top node anymore)
G.add_edges_from([(vname,-u) for u in B[cv] if u!=v])
return G
[docs]def graph_clique_number(G,cliques=None):
"""Return the clique number (size of the largest clique) for G.
An optional list of cliques can be input if already computed.
"""
if cliques is None:
cliques=find_cliques(G)
return max( [len(c) for c in cliques] )
[docs]def graph_number_of_cliques(G,cliques=None):
"""Returns the number of maximal cliques in G.
An optional list of cliques can be input if already computed.
"""
if cliques is None:
cliques=list(find_cliques(G))
return len(cliques)
[docs]def node_clique_number(G,nodes=None,cliques=None):
""" Returns the size of the largest maximal clique containing
each given node.
Returns a single or list depending on input nodes.
Optional list of cliques can be input if already computed.
"""
if cliques is None:
if nodes is not None:
# Use ego_graph to decrease size of graph
if isinstance(nodes,list):
d={}
for n in nodes:
H=networkx.ego_graph(G,n)
d[n]=max( (len(c) for c in find_cliques(H)) )
else:
H=networkx.ego_graph(G,nodes)
d=max( (len(c) for c in find_cliques(H)) )
return d
# nodes is None--find all cliques
cliques=list(find_cliques(G))
if nodes is None:
nodes=G.nodes() # none, get entire graph
if not isinstance(nodes, list): # check for a list
v=nodes
# assume it is a single value
d=max([len(c) for c in cliques if v in c])
else:
d={}
for v in nodes:
d[v]=max([len(c) for c in cliques if v in c])
return d
# if nodes is None: # none, use entire graph
# nodes=G.nodes()
# elif not isinstance(nodes, list): # check for a list
# nodes=[nodes] # assume it is a single value
# if cliques is None:
# cliques=list(find_cliques(G))
# d={}
# for v in nodes:
# d[v]=max([len(c) for c in cliques if v in c])
# if nodes in G:
# return d[v] #return single value
# return d
[docs]def number_of_cliques(G,nodes=None,cliques=None):
"""Returns the number of maximal cliques for each node.
Returns a single or list depending on input nodes.
Optional list of cliques can be input if already computed.
"""
if cliques is None:
cliques=list(find_cliques(G))
if nodes is None:
nodes=G.nodes() # none, get entire graph
if not isinstance(nodes, list): # check for a list
v=nodes
# assume it is a single value
numcliq=len([1 for c in cliques if v in c])
else:
numcliq={}
for v in nodes:
numcliq[v]=len([1 for c in cliques if v in c])
return numcliq
[docs]def cliques_containing_node(G,nodes=None,cliques=None):
"""Returns a list of cliques containing the given node.
Returns a single list or list of lists depending on input nodes.
Optional list of cliques can be input if already computed.
"""
if cliques is None:
cliques=list(find_cliques(G))
if nodes is None:
nodes=G.nodes() # none, get entire graph
if not isinstance(nodes, list): # check for a list
v=nodes
# assume it is a single value
vcliques=[c for c in cliques if v in c]
else:
vcliques={}
for v in nodes:
vcliques[v]=[c for c in cliques if v in c]
return vcliques